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Minimum Variance Portfolio Composition
Abstract
Empirical studies document that equity portfolios constructed to have the lowest possible risk have surprisingly high average returns. We derive an analytic solution for the long-only minimum variance portfolio under the assumption of a single-factor covariance matrix. The equation for optimal security weights has a simple and intuitive form that provides several insights on minimum variance portfolio composition. While high idiosyncratic risk can lead to a low security weight, high systematic risk takes the large majority of investable securities out of long-only solutions. The relatively small set of securities that remain have market betas below an analytically specified threshold beta. The math also shows that the ratio of portfolio beta to the threshold beta dictates the portion of ex-ante portfolio variance that is market-factor related. We verify and illustrate the portfolio mathematics using historical data on the U.S. equity market and explore how the single-factor analytic results compare to numerical optimization under a generalized covariance matrix. The analytic and empirical results of this study suggest that minimum variance portfolio performance is largely a function of the long-standing empirical critique of the traditional CAPM that low beta stocks have relatively high average returns.
... While the largest eigenvalue typically exceeds the level of uctuations by two orders of magnitude, the other eigenvalues rapidly reach this level and become non-informative. Several researchers employed the random matrix theory to distinguish economically signicant eigenvalues from noise [8,9,10,11,12]. In particular, Laloux et al. showed that only 6% of the eigenvalues carried some information of the S&P 500 (1991)(1992)(1993)(1994)(1995)(1996), while the remaining 94% eigenvalues were hidden by noise [8]. Guhr and Kalber proposed an alternative statistical approach to reduce noise that they called power mapping [13]. ...
... It has the unique property that optimal security weights are solely dependent on security covariance matrix without regards to the expected returns; hence, it does not rely on any specic expected return estimate (see e.g. [12]), which makes it appear more robust than the mean-variance framework. Minimum-Variance strategies have gained popularity notably due to the empirical nding that low-volatility stocks tend to have returns that tend to exceed in average the market returns [13]. ...
... is the conditional variance of the factor return. In deriving this result we used the rst property of the Two-Factor model (12) and the relation ...
Une nouvelle méthode a été mise en place pour débruiter la matrice de corrélation des rendements des actions en se basant sur une analyse par composante principale sous contrainte enexploitant les données financières. Des portefeuilles, nommés "Fundamental Maximum variance portfolios", sont construits pour capturer de manière optimale un style de risque défini par un critère financier ("Book", "Capitalization",etc.). Les vecteurs propres sous contraintes de la matrice de corrélation, qui sont des combinaisons linéaires de ces portefeuilles, sont alors étudiés. Grâce à cette méthode, plusieurs faits stylisés de la matrice ont été mis en évidence dont: i) l'augmentation des premières valeurs propres avec l'échelle de temps de 1 minute à plusieurs mois semble suivre la même loi pour toutes les valeurs propres significatives avec deux régimes; ii) une loi _universelle_ semble gouverner la composition de tous les portefeuilles "Maximum variance". Ainsi selon cette loi, les poids optimaux seraient directement proportionnels au classement selon le critère financier étudié; iii) la volatilité de la volatilité des portefeuilles "Maximum Variance_" qui ne sont pas orthogonaux, su_rait à expliquer une grande partie de la diffusion de la matrice de corrélation; iv) l'effet de levier (augmentation de la première valeur propre avec la baisse du marché) n'existe que pour le premier mode et ne se généralise pas aux autres facteurs de risque. L'effet de levier sur les beta, sensibilité des actions avec le "market mode", rend les poids du premier vecteur propre variables.
... ent. Many other studies specific to U.S. markets including (Chan, Karceski, & Lakonishok, 1999), (Schawartz, 2000), (Jagannathan & Ma, 2003) report both higher returns and lower realized risks for the minimum variance portfolio (MVP) versus a capitalization-weighted benchmark (MWP). We discuss some of the major studies and their propositions below. Clarke et al. (Clarke, DeSilva, & Thorley, 2006a) study focus on the characteristics of minimum-variance (MV) portfolios. The study reports that MV portfolios based on the 1,000 largest U.S. stocks over the period of 1968 – 2005, achieved a volatility reduction of about 25% while delivering comparable or even higher average returns than the market portfolio. This means that the MV por ...
... Given below are some of the examples of different methodological choices used in some of the important studies. (Clarke, DeSilva, & Thorley, 2006a), (Chan, Karceski, & Lakonishok, 1999), (Schawartz, 2000), (Jagannathan & Ma, 2003) use minimum variance based approach whereas (Blitz & Vliet, 2007), (Ang, Hodrick, Xing, & Zhang, 2006), (Baker & Haugen, 2012), (Frazzini & Pedersen, 2014) use risk measure based sorting approach. ...
... (Clarke, DeSilva, & Thorley, 2006a), (Bali & Cakici, 2008), (Baker, Bradley, & Wurgler, 2011) use standard deviation or variance as risk measure, (Frazzini & Pedersen, 2014) use beta as risk measure. (Blitz & Vliet, 2007) use both standard deviation as well as beta as risk measure. ...
Number of studies show that portfolio of low risk stocks outperform portfolio of high risk stocks as well as the market portfolio over the full market cycle on risk adjusted basis and in some cases, absolute basis as well in some cases. This surprising contradiction to classic finance theory led by CAPM has held its ground over long periods of time, across different markets and different methodological choices. This review paper aims at contributing to the body of knowledge in four ways. One, it highlights and links different strands of literature on low risk anomaly that has evolved over a period of time. Second, it highlights, different methodological choices that have been used. Third, it classifies explanations for persistence of risk anomaly in to economic and behavioral explanations and explanations that try to explain the anomaly away. Fourth, it reviews the state of current research and explores potential but yet underexplored areas of research on risk anomaly.
... Unfortunately, it is well known that optimal Minimum Variance portfolios based on the sample covariance matrix yield a large proportion of negative weights (we refer to Clarke et al. 2011 for more details on this subject). Even after elaborate enhancements of the covariance matrix, many weights remain negative. ...
... DeMiguel et al. (2009) argue that " shortsaleconstrained minimum-variance portfolios [. . . ] tend to assign a weight different from zero to only a few of the assets " , while it is noted by Clarke et al. (2011) that their long-only minimum variance " portfolio averages about 120 long securities, i.e., about 12 % of the 1000-security investable set " . The second way to deal with Long-Short Minimum Variance weights is to introduce norm constraints on the weights. 2 This introduces a trade-off between losing optimality on the variance of the portfolio while at the same time gaining diversification. ...
... Only Minimum Variance portfolio (Jagannathan and Ma 2003; Clarke et al. 2011) minimizes volatility, but is often poorly diversified. To the best of our knowledge, there is no weighting scheme in the literature which aims at combining those three features and the present article is meant to fill this void. ...
We build on a one parameter family of weighting schemes arising from \(L^2\) -constrained portfolio optimization problems. The parameter allows to fine tune the trade-off between the volatility and the diversification of the portfolio. We propose two criteria in order to determine two unique portfolios: the first criterion requires that no weights be negative while the second one imposes a target diversification which is median between full concentration and full diversification. Both portfolios are empirically compared to classical benchmarks. The first one behaves very much like other popular Long-Only weighting schemes while the second displays a more aggressive profile, while generating moderate turnover. We also discuss implementation issues, as well as estimation related problems.
... To evaluate the performance of the model in terms of return and risk the studies by Polson and Tew (2000) and Clarke et al (2011) were used as references. The objective was to analyze the performance of portfolios with the market indicator and the classical model MV model, and tries to improve them. ...
... The analysis was prepared simultaneously with long-term investment portfolios in the same period as the Bovespa index, i.e. quarterly, using both the BMM and the classic MV.Table 01 of the portfolios produced by each model, compared with the market index. The required return for the construction of the portfolios was 0.00%, ie the portfolios of both models were executed using the minimum variance, in line with Jobson and Korkie (1980) and Clarke et al (2006 Clarke et al ( , 2011) who suggest doing so, especially because it does not depend on asset returns. As shown inTable 01, the cumulative return of 510.10% and the average daily return of 0.10% was well above the BMM Ibovespa and the MV model. ...
... In general, the rate of Sharpe (1966) used as a comparison showed that the model becomes satisfactory, in return and risk, especially in longer maturities. These results corroborate the traditional concept that one can take a little more risk in exchange for higher returns, found by Boyle and Tian (2007) and refutes the idea that to get higher returns, it is not necessary to assume greater risk, as in Clarke et al (2006 Clarke et al ( , 2011). Another important result was to reduce the number of assets in the portfolios, there being an average of five, and the gain obtained in return and risk from the model contributes to the inquiry undertaken by Black and Litterman (1992), about what diversification really is. ...
Classical Mean-Variance (MV) model has been criticized and it is observed that the use of covariance has sensitivity to the parameter estimation. Thus, the present study set out to formulate a new matrix model, using Bayesian inference as a way to replace covariance in MV, called BMM – Bayesian Matrix Model. To evaluate the model, some hypotheses were analyzed using an ex post facto method and sensitivity analysis. The returns earned from May 2002 to December 2009 demonstrated the superiority of BMM in relation to the MV model and the Bovespa Index, but by taking more diversifiable risks than MV does.
... where R i (t) is the return of asset i, R mkt (t) is the return of the market factor, ε i (t) ∼ N (0,σ 2 i ) is the idiosyncratic risk andσ i is the idiosyncratic volatility. Clarke et al. (2011) andScherer (2011) showed that: ...
... Therefore, we note that the minimum variance portfolio is exposed to stocks with low specific volatilityσ i and low beta β mkt,i . More precisely, if asset i has a market beta β mkt,i smaller than the threshold β mkt , the weight of this asset is positive: x i > 0. If β mkt,i > β mkt , then x i < 0. Clarke et al. (2011) extended Formula (7) to the long-only case, where β mkt is another threshold. ...
Like ESG investing, climate change is an important concern for asset managers and owners, and a new challenge for portfolio construction. Until now, investors have mainly measured carbon risk using fundamental approaches, such as with carbon intensity metrics. Nevertheless, it has not been proven that asset prices are directly impacted by these fundamental-based measures. In this paper, the authors focus on another approach, which consists in measuring the sensitivity of stock prices with respect to a carbon risk factor. In the opinion of the authors, carbon betas are market-based measures that are complementary to carbon intensities or fundamental-based measures when managing investment portfolios, because carbon betas may be viewed as an extension or forward-looking measure of the current carbon footprint. In particular, they show how this new metric can be used to build minimum variance strategies and how they impact their portfolio construction.
... where R i (t) is the return of asset i, R mkt (t) is the return of the market factor, ε i (t) ∼ N 0,σ 2 i is the idiosyncratic risk andσ i is the idiosyncratic volatility. Clarke et al. (2011) andScherer (2011) showed that: ...
... Therefore, we note that the minimum variance portfolio is exposed to stocks with low specific volatilitỹ σ i and low beta β mkt,i . More precisely, if asset i has a market beta β mkt,i smaller than the threshold β mkt , the weight of this asset is positive: Clarke et al. (2011) extended Formula (7) to the long-only case, where β mkt is another threshold. In this case, if β mkt,i < β mkt , x i > 0 and if β mkt,i ≥ β mkt , x i = 0. We consider an extension of the CAPM by including the BMG risk factor: ...
Like ESG investing, climate change is an important concern for asset managers and owners, and a new challenge for portfolio construction. Until now, investors have mainly measured carbon risk using fundamental approaches, such as with carbon intensity metrics. Nevertheless, it has not been proven that asset prices are directly impacted by these fundamental-based measures. In this paper, we focus on another approach, which consists in measuring the sensitivity of stock prices with respect to a carbon risk factor. In our opinion, carbon betas are market-based measures that are complementary to carbon intensities or fundamental-based measures when managing investment portfolios, because carbon betas may be viewed as an extension or forward-looking measure of the current carbon footprint. In particular, we show how this new metric can be used to build minimum variance strategies and how they impact their portfolio construction.
... where R i (t) is the return of asset i, R mkt (t) is the return of the market factor, ε i (t) ∼ N 0,σ 2 i is the idiosyncratic risk andσ i is the idiosyncratic volatility. Clarke et al. (2011) andScherer (2011) showed that: ...
... Therefore, we note that the minimum variance portfolio is exposed to stocks with low specific volatilitỹ σ i and low beta β mkt,i . More precisely, if asset i has a market beta β mkt,i smaller than the threshold β mkt , the weight of this asset is positive: Clarke et al. (2011) extended Formula (7) to the long-only case, where β mkt is another threshold. In this case, if β mkt,i < β mkt , x i > 0 and if β mkt,i ≥ β mkt , x i = 0. We consider an extension of the CAPM by including the BMG risk factor: ...
Like ESG investing, climate change is an important concern for asset managers and owners, and a new challenge for portfolio construction. Until now, investors have mainly measured carbon risk using fundamental approaches, such as with carbon intensity metrics. Nevertheless, it has not been proven that asset prices are directly impacted by these fundamental-based measures. In this paper, we focus on another approach, which consists in measuring the sensitivity of stock prices with respect to a carbon risk factor. In our opinion, carbon betas are market-based measures that are complementary to carbon intensities or fundamental-based measures when managing investment portfolios, because carbon betas may be viewed as an extension or forward-looking measure of the current carbon footprint. In particular, we show how this new metric can be used to build minimum variance strategies and how they impact their portfolio construction.
... In portfolio mathematics, constrained solutions are usually intractable, but assuming a single-factor risk model provides for a straightforward and intuitive expression for the best weights. The study compared single factor analytics to quantitative optimizations on a very generalized covariance matrix and discovered that accounting for non-market sources of security correlation hardly slightly changes the analytically obtained optimal weights [9]. Kan and Zhou calculated analytically the expected loss function associated with employing sample means and the covariance matrix of returns to determine the best portfolio. ...
... The extension to the long-only case follows the semi-formal proof formulated by Clarke et al. (2011). We note I u = {1, . . . ...
This article studies the impact of carbon risk on stock pricing. To address this, we consider the seminal approach of Görgen et al. (2019), who proposed estimating the carbon financial risk of equities by their carbon beta. To achieve this, the primary task is to develop a brown-minus-green (or BMG) risk factor, similar to Fama and French (1992). Secondly, we must estimate the carbon beta using a multi-factor model. While Görgen et al. (2019) considered that the carbon beta is constant, we propose a time-varying estimation model to assess the dynamics of the carbon risk. Moreover, we test several specifications of the BMG factor to understand which climate change-related dimensions are priced in by the stock market. In the second part of the article, we focus on the carbon risk management of investment portfolios. First, we analyze how carbon risk impacts the construction of a minimum variance portfolio. As the goal of this portfolio is to reduce unrewarded financial risks of an investment, incorporating the carbon risk into this approach fulfils this objective. Second, we propose a new framework for building enhanced index portfolios with a lower exposure to carbon risk than capitalization-weighted stock indices. Finally, we explore how carbon sensitivities can improve the robustness of factor investing portfolios.
... In particular, they depend on the relative performance of the EW and MV portfolios with respect to the CW portfolio. 9 According to Johansson and Pekkala (2013), the investment capacity ratio of asset i corresponds to the CW weight x cw,i divided by the portfolio weight x i (see also NBIM, 2012): ...
In this article, we consider smart beta indexing, which is an alternative to capitalization-weighted (CW) indexing. In particular, we focus on risk-based (RB) indexing, the aim of which is to capture the equity risk premium more effectively. To achieve this, portfolios are built which are more diversified and less volatile than CW portfolios. However, RB portfolios are less liquid than CW portfolios by construction. Moreover,
they also present two risks in terms of passive management: tracking difference risk and tracking error risk. Smart beta investors then have to a puzzle out the trade-off between diversification, volatility, liquidity and tracking error. This article examines the trade-off relationships. It also defines the return components of smart beta indexes.
... All market and value betas are positive and significant. The minimum-variance benchmark has the smallest market beta, which supports the results of Scherer (2011) and Clarke, de Silva, and Thorley (2006Thorley ( , 2011): the minimum-variance portfolio loads more on smaller market risk stocks, and has higher one-factor alpha than value-weighted portfolio. In the four-factor model the minimum-variance portfolio has a less negative size beta, which means that this benchmark loads less on big-cap stocks if compared with a value-weighted portfolio. ...
We compare the performance of a wide variety of benchmarks: traditional, fundamentals-based and optimization-based. We find that for a set of all stocks of the S&P500 index during the period from February 1989 to De-cember 2011 traditional and new benchmark portfolios perform similarly according to a variety of return, risk, turnover, and diversification performance metrics. Moreover, the difference between traditional value-or equal-weighted benchmark and new benchmark portfolios is not statistically significant. We identify a set of basis benchmarks, which span both the set of new and the set of traditional benchmarks. The first basis benchmark explains three quarters of the variation of all benchmark portfolios; correlation between this basis benchmark and systematic market factor is 96% for the last 10 years period. We conclude that the strongest driving force of all considered benchmark portfolios is the market factor. Irrespective of the benchmark portfolio, managers mainly track the market and do it in statistically sufficient way during the last 23 years. The difference in the performance of various benchmarks can be attributed to the skill to outperform the market. In the long run these skills are washed out. Our work has implications for big mutual, pension and hedge funds with fairly big number of stocks in their portfolios and long investment time horizon. For these funds the choice of the benchmark is not important.
... Evidence on a flatter systematic risk and return relation than expected as per the CAPM comes from , and Fama and French (1992). Further, Haugen and Heins (1975), Haugen and Baker (1991), Haugen and Baker (1996), Clarke et al. (2006, Blitz and Vliet (2005) and Frazzini and Pedersen (2014) offer evidence on the negative relation between risk and return. Among others, Choueifaty and Coignard (2008), Soe (2012), Carvalho et al. (2012also find evidence for low-risk anomaly. ...
We offer empirical evidence that stocks with low volatility earn higher risk-adjusted returns compared to high volatility stocks in the Indian stock market. The annualised excess returns for the low and high volatility decile portfolios amount to 11.40% and 1.30%, respectively, over the period January 2001 to June 2015. The difference of returns is statistically and economically significant for both low and high-risk stocks. Using risk measures of standard deviation and beta, the volatility effect remains after controlling for size, value and momentum. We uncover that the volatility effect is not statistically significant after controlling for beta effect. Our evidence for volatility effect is not dominated by small and illiquid stocks. Our results show that the low volatility portfolio outperforms benchmark portfolio not only in down market but also in up market conditions.
... A major disadvantage of such methods is that they are hard to calculate due to the high dimensionality of the covariance matrix that is required before stock weights can be calculated for this index. Though, based on empirical studies, it has been seen that the minimum variance index outperforms the market capitalisation index in a falling market, but lags in a rising market (Haugen and Baker 1991; Clarke, de Silva, and Thorley 2006; Chia et al. 2011). On the other hand, Choueifaty and Coignard (2008) developed a risk weighted method that utilises the Sharpe ratio in order to find the weight of these stocks within the index. ...
Major stock indexes are developed on the market capitalization or price weighted indexation method. The Australian Stock Exchange 50 (ASX50) index is a market capitalization index of the top 50 Australian stocks. Fundamental indexation, equal weighted index and risk weighted index methods have recently been developed as an alternative to the market cap and price indexes. However, empirical studies do not conclusively prove if these alternate methods are more efficient to the existing market cap or price weighted methods. Also, the fundamental index method provides a higher alpha, while the risk weighted index methods focus on risk reduction through diversification. There is a gap to develop another passive indexation method in order to provide the investor a higher return (alpha) and lower volatility. This paper re-weights the ASX50 index using the risk weighted alpha method and provides higher weight to stocks that have increasing returns and lower volatility. The empirical study for ASX50 index from 2002-2012 is undertaken and results show that the risk weighted alpha method provides higher return and has lower systematic risk than the ASX50 index.
... We now consider an alternative performance-based model combination rule based on the minimum variance portfolio policy. A very large body of literature in portfolio optimization considers this particular policy; see, for instance, Clarke et al. (2006 Clarke et al. ( , 2011) for extensive practitioner-oriented studies on the performance and on the composition of minimum variance portfolios. This policy can be seen as a particular case of the traditional mean-variance optimization. ...
We devise a novel approach to combine predictions of high dimensional conditional covariance matrices using economic criteria based on portfolio selection. The combination scheme takes into account not only the portfolio objective function but also the portfolio characteristics in order to define the mixing weights. Three important advantages are that i) it does not require a proxy for the latent conditional covariance matrix, ii) it does not require optimization of the combination weights, and iii) it holds the equally-weighted model combination as a particular case. Empirical applications involving three large data sets from different markets show that the proposed economic-based combinations of multivariate GARCH forecasts leads to mean-variance portfolios with higher risk-adjusted performance in terms of Sharpe ratio as well as to minimum variance portfolios with lower risk on an out-of-sample basis with respect to a number of benchmark specifications.
... In this context , an interest in minimum variance portfolios has greatly increased, the optimization of which does not require the prognosis of the expected returns and is limited to the covariance matrix only. Unfortunately, minimum variance portfolios also do not avoid the high concentration of low variance securities (Clarke et al. 2011). Most of the proposed MV optimization solution techniques, in one way or another, are related to constraints on model input parameters or portfolio weights (Frost, Savarino 1988; Jagannathan, Ma 2003 ). ...
There is little literature considering effects that the loss-gain threshold used for dividing good and bad outcomes by all downside (upside) risk measures has on portfolio optimization and performance. The purpose of this study is to assess the performance of portfolios optimized with respect to the Omega function developed by Keating and Shadwick at different levels of the threshold returns. The most common choices of the threshold values used in various Omega studies cover the risk-free rate and the average market return or simply a zero return, even though the inventors of this measure for risk warn that “using the values of the Omega function at particular points can be critically misleading” and that “only the entire Omega function contains information on distribution”. The obtained results demonstrate the importance of the selected values of the threshold return on portfolio performance – higher levels of the threshold lead to an increase in portfolio returns, albeit at the expense of a higher risk. In fact, within a certain threshold interval, Omega-optimized portfolios achieved the highest net return, compared with all other strategies for portfolio optimization using three different test datasets. However, beyond a certain limit, high threshold values will actually start hurting portfolio performance while meta-heuristic optimizers typically are able to produce a solution at any level of the threshold, and the obtained results would most likely be financially meaningless.
... In particular, they depend on the relative performance of the EW and MV portfolios with respect to the CW portfolio. 9 According to Johansson and Pekkala (2013), the investment capacity ratio of asset i corresponds to the CW weight x cw,i divided by the portfolio weight x i (see also NBIM, 2012): ...
In this article, we consider smart beta indexing, which is an alternative to capitalization-weighted (CW) indexing. In particular, we focus on risk-based (RB) indexing, the aim of which is to capture the equity risk premium more effectively. To achieve this, portfolios are built which are more diversified and less volatile than CW portfolios. However, RB portfolios are less liquid than CW portfolios by construction. Moreover, they also present two risks in terms of passive management: tracking difference risk and tracking error risk. Smart beta investors then have to a puzzle out the trade-off between diversification, volatility, liquidity and tracking error. This article examines the trade-off relationships. It also defines the return components of smart beta indexes.
... Atsižvelgiant į tai, smarkiai išaugo susidomėji mas minimalios dispersijos portfeliais, kuriems optimizuoti nereikalinga tikėtinųjų grąžų prognozė, o užtenka tik ko variacinės matricos. Deja, minimalios dispersijos portfeliai taip pat neišvengia didelės vertybinių popierių, pasižymin čių nedidele dispersija, koncentracijos (Clarke et al. 2011) Dauguma pasiūlytų MV optimizavimo problemos sprendimo būdų vienaip ar kitaip yra susiję su mode lio įvesties parametrų arba portfelio svorių apribojimais (Frost, Savarino 1988; Jagannathan, Ma 2003). Akivaizdu, kad, ribojant maksimalius svorius, užkertamas kelias nelogiškai didelei akcijų koncentracijai ir užtikrinamas geresnis rizikos išskaidymas, tačiau svorių apribojimai reiškia, kad mažiau remiamasi optimizavimo procedūro mis ir rinkos " signalų " patikimumu, o daugiau – " naiviu " rizikos valdymu. ...
This paper considers portfolio optimisation with respect to the Omega function, proposed by Keating & Shadwick, and investigates its empirical performance compared with the traditional approaches for portfolio optimisation. The results were inline with expectations: omegaoptimised portfolio surpassed the performance of equally weighted portfolio and meanvariance portfolios based on sample estimates and was inferior only to, or equalled portfolios based on regularised estimates of the covariance matrix. In addition, it can be stated that optimisation of the Omega function was based on historical returns used as “scenarios”, so it can be reasonably expected to receive better results by using inputs or scenarios obtained by more sophisticated methods.
Omega atžvilgiu optimizuoto akcijų portfolio empiriniai tyrimai
Santrauka
Straipsnyje nagrinėjamas portfelio optimizavimas omega funkcijos, pasiūlytos Keating ir Shadwick, atžvilgiu naudojant empirinius duomenis ir lyginant gautus rezultatus su rezultatais, gautais taikant tradicinius portfelio optimizavimo metodus. Gauti rezultatai iš esmės atitiko lūkesčius – omega atžvilgiu optimizuotas portfelis pralenkė vienodų svorių bei vidurkių ir dispersijos atžvilgiu optimizuotus portfelius, kai šių portfelių optimizavimas buvo grindžiamas įprastais imties įverčiais ir atsiliko ar prilygo portfeliams, sudarytiems remiantis suderintais kovariacinės matricos įverčiais. Be to, pažymėtina, kad, optimizuojant omega funkciją, naudotas pats paprasčiausias „istorinis“ scenarijus, todėl galima pagrįstai tikėtis, kad, naudojant sudėtingesnius scenarijų sudarymo algoritmus, galima gauti geresnių rezultatų.
Reikšminiai žodžiai: portfelio optimizavimas, omega funkcija.
... where b is the beta threshold (see also Clarke, 2011). When we in addition fully shrink the factor V of volatility and the factor C of correlation TopX (out of N) assets on the general rank function: ...
Modern Portfolio Theory (MPT), as developed by Markowitz (1952) and others, is often described as a nice but impractical theory. The full MPT framework is very sensitive to parameters like the expected returns which are estimated with errors, resulting in allocations with even larger errors. This is known as the multiplication of errors. The same holds for the expected covariance matrix. In this paper we combine MPT with momentum, a simple covariance model and shrinkage estimators. First, we use historical estimates based on short (up to one year) lookback periods, in contrast to the traditional (multi-year) approach. Second, we use the “single-index model” of Elton (1976) to structure the covariance matrix and to arrive at an elegant analytical formula for the optimal allocations. Finally, we reduce estimation errors by partially “shrinking” all estimates for expected returns, volatilities and cross-correlations towards their means over assets. We call the resulting “tactical” (short-term) MPT model the “Modern Asset Allocation” (MAA). We illustrate the MAA model on nine universes (with 7 to 130 assets) over 1997-2013 and show that the MAA model beats the simple EW model consistently, proving the usefulness of MPT.
... Some of the common elements of risk-based investment strategies are elucidated in Clarke et al (2011), who provided a reexpression of (3.1) in a market with a single risk factor: (4.1) shows that the weight of asset n decreases as either its CAPM market model beta, ˇ n , or its idiosyncratic variance, 2 i , increases. Furthermore,Figure 1 on the facing page shows cumulative returns to the three low-risk strategies and the balanced portfolio. ...
Risk-only investment strategies have been growing in popularity as traditional investment strategies have fallen short of return targets over the last decade. However, risk-based investors should be aware of four things. First, theoretical considerations and empirical studies show that apparently distinct risk-based investment strategies are manifestations of a single effect. Second, turnover and associated transaction costs can be a substantial drag on return. Third, capital diversification benefits may be reduced. Fourth, there is an apparent connec-tion between performance and risk diversification. To analyze risk diversification benefits consistently, we introduce the risk diversification index, which measures risk concentrations and complements the Herfindahl–Hirschman index for capital concentrations.
... First introduced by Markowitz (1952), this method involves solving the basic quadratic optimization problem using a single linear constraint in equality. 3 Its popularity has grown ever since, with studies as recent as Clarke et al (2011) and Scherer (2010). P observations are assumed to be iid normal. ...
We divide hedging methods between single-period and multi-period. After reviewing some well-known hedging algorithms, two new procedures are introduced, called Dickey-Fuller Optimal (DFO), Mini-Max Subset Correlation (MMSC). The former is a multi-period, cointegration-based hedging method that estimates the holdings that are most likely to deliver a hedging error absent of unit root. The latter is a single-period method that studies the geometry of the hedging errors and estimates a hedging vector such that subsets of its components are as orthogonal as possible to the error. We test each method for stability and robustness of the derived hedged portfolio. Results indicate that DFO produces estimates similar to the Error Correction Method, but more stable. Likewise, MMSC estimates are similar to Principal Component Analysis but more stable. Finally, a generalized Box-Tiao Canonical Decomposition (BTCD) method is proposed, which is of the multi-period class. BTCD estimates are also very stable, and cannot be related to any of the aforementioned methodologies. Finally, we find that all three advanced hedging methods (MMSC, BTCD, DFO) perform well.
Optimizing portfolio has been a popular topic since it was proposed because it can reduce the investment risk. This study selected five active stocks from different industries, including Online E-Commerce, Commercial Banks, Motor Vehicles, Mobile Communication Production, and Telecommunications. Then they were allocated into five kinds of portfolios, which are tangency portfolios and minimum variance portfolios under Capital Asset Pricing Model and Fama-French three-factor Model, as well as the 1/N portfolio. The results found that ‘BAC’ and ‘T’ have the largest weight and the lowest weight respectively in the two models. The comparison of the five portfolios showed that the portfolio with the highest cumulative return is the tangency portfolio under FF3F Model, and the 1/N portfolio also performed well. Only these two portfolios outperformed the SPDR S&P 500EFT Trust. This research may help investors focused on the five sectors mentioned above to have a better idea of how to allocate their capital.
A new methodology has been introduced to clean the correlation matrix of single stocks returns based on a constrained principal component analysis using financial data. Portfolios were introduced, namely "Fundamental Maximum Variance Portfolios", to capture in an optimal way the risks defined by financial criteria ("Book", "Capitalization", etc.). The constrained eigenvectors of the correlation matrix, which are the linear combination of these portfolios, are then analyzed. Thanks to this methodology, several stylized patterns of the matrix were identified: i) the increase of the first eigenvalue with a time scale from 1 minute to several months seems to follow the same law for all the significant eigenvalues with 2 regimes; ii) a universal law seems to govern the weights of all the "Maximum variance" portfolios, so according to that law, the optimal weights should be proportional to the ranking based on the financial studied criteria; iii) the volatility of the volatility of the "Maximum Variance" portfolios, which are not orthogonal, could be enough to explain a large part of the diffusion of the correlation matrix; iv) the leverage effect (increase of the first eigenvalue with the decline of the stock market) occurs only for the first mode and cannot be generalized for other factors of risk. The leverage effect on the beta, which is the sensitivity of stocks with the market mode, makes variable the weights of the first eigenvector.
We explore futures hedging based on the global minimum variance strategy. As evidenced by using eleven of the world’s major stock market indexes and their corresponding futures contracts, the results show that the global minimum variance hedge may deviate statistically from the Ederington (J Finance 43(1):157–170, 1979) minimum variance hedge. We also present a regression approach to testing the hedge ratios and futures positions when the noise terms follow a normal distribution. In the illustration examined, we show that the global minimum variance hedge provides a more economically significant information ratio yield than that under the minimum variance hedge.
The capital asset pricing model (CAPM) developed by Sharpe (1964) is the starting point for the arbitrage pricing theory (APT). It uses a single risk factor to model the risk premium of an asset class. However, the CAPM has been the subject of important research, which has highlighted numerous empirical contradictions. Based on the
APT theory proposed by Ross (1976), Fama and French (1992) and Carhart (1997) introduce other common factors models to capture new risk premia. For instance, they consequently define equity risk factors, such as market, value, size and momentum. In recent years, a new framework based on this literature has emerged to define strategic asset allocation. Similarly, index providers and asset managers now offer the opportunity to invest in these risk factors through factor indexes and mutual funds. These two approaches led to a new paradigm called `factor investing' (Ang, 2014). Factor investing seems to solve some of the portfolio management issues that emerged in the past, in particular for long-term investors. However, some questions arise, especially with the number of risk factors growing over the last few years (Cochrane, 2011). What
is a risk factor? Are all risk factors well-rewarded? What is their level of stability and robustness? How should we allocate between them? The main purpose of this paper is to understand and analyze the factor investing approach in order to answer these questions.
In this study, we show that patterns in returns behave as if investors, influenced by their level of optimism, selected stocks according to their volatility. Our goal is to confirm the contribution of behavioral finance while showing that investor sentiment can be profitably used by practitioners. We incorporate volatility in the relationship between investor sentiment and future returns, this is the main originality of our approach. Our methodology consists in comparing returns, volatility and higher-order moments of portfolios managed with investor sentiment against those obtained either with passive (buy and hold) portfolio management or with a minimum variance portfolio. Portfolios managed with investor sentiment have better returns and involve less risk under certain conditions.
This paper studies the out of sample risk reduction of global minimum variance portfolio. The analysis are drown from the discussions of Jagannathan and Ma (2003) regarding the risk reduction in US stock portfolios using portfolio constraints. We estimate the covariance matrix using the sample covariance matrix approach and derive optimal minimum variance portfolios considering upper/lower bounds and no restrictions. Results are shown under different revision frequency and transaction costs assumed. The data used are monthly indices of stocks, bonds, gold oil and spreads from 1996 until 2013. Unconstrained GMVPs result in the lowest out of sample variance, while unconstrained GMVPs of global bond portfolios performs the best in terms of risk reduction. Diversification through global asset classes result in a better strategy than international stock diversification regarding risk, as suggested by the literature.
Our paper investigates the efficiency of portfolio diversification at Romanian stock market. In order to do this, we use data mining techniques to study the effect that the number of assets has on the total portfolio variance. We use the liquidity of transactions during Jan. 1st 2011 and May 11th 2012 to rank 42 of the most active stocks traded at Bucharest Stock Exchange, and after that we simulate portfolios and compute the Markowitz efficient frontier for combinations of assets that start with only the most liquid company and continue until all the 42 stocks are included. Our results show that portfolio risk decreases as the number of assets increases, both for the minimum variance portfolio (MVP) and for other discretionary types of portfolios. We also find that the marginal benefits of diversification are high at the earlier stages and decreases as the number of assets increase. We continue by simulating different combinations between the 42 stocks universe of risky portfolios and the risk free asset, and we confirm that best choice of capital market line (CML) portfolios, for all levels of risk aversion, are found when the number of assets in the portfolio is higher. © Iulian Panait, Ecaterina Oana Slavescu, Alexandru Constantinescu, 2013.
The cumulative return to a levered strategy is determined by five elements that fit together in a simple and useful formula. A previously undocumented element is the covariance between leverage and excess return to the fully invested source portfolio underlying the strategy. In an empirical study of volatility-targeting strategies over the 84-year period 1929-2013, this covariance accounted for a reduction in return that substantially diminished the Sharpe ratio in all cases.
In this paper we develop a new approach to portfolio construction that relies solely on the covariance structure of the investment opportunity set. Using this new approach leads to an alternative to the mean–variance efficient set of portfolios as traditionally implemented. We define an “optimally diversified” portfolio as one which is equally correlated to each of its components. We show empirically in this paper that using this “optimally diversified” portfolio construction methodology results in a significant improvement in portfolio performance over traditional mean–variance efficient portfolios as well as naively diversified (e.g., equally weighted) portfolios.
In this paper we will try to improve on the Modern Portfolio Theory (MPT) as developed by Markowitz (1952). As a first step, we combine the MPT model with generalized momentum (see Keller 2012) in order to arrive at a "tactical" MPT. In our second step, we will use the single index model (Elton, 1976) to arrive at an analytical solution for a long-only maximum Sharpe allocation. We will call this the MAA model, for Modern Asset Allocation. In our third step, we use shrinkage estimators in our formula for asset returns, volatilities and correlations to arrive at practical allocations. In addition, as a special cases, we arrive at EW (Equal Weight), Minimum Variance (MV), Maximum Diversification (MD) and (naïve) Risk Parity (RP) submodels of MAA. These EW, MV, MD and RP models are sometimes called "smart-beta" models.
This article introduces an algorithm for tail risk hedging and compares it to other existing methods. This algorithm adjusts the exposure level based on a measure of tail risk obtained by applying Extreme Value Theory (EVT) to estimate Conditional Value at Risk (CVaR). This method is applied to the S&P 500 and MSCI Emerging Markets equity indexes between 2000 and 2012 and its performance is compared to cash and options based tail hedging strategies. The cash based methods are shown to significantly increase risk adjusted returns and reduce drawdowns, while the options based strategy suffers a decrease in performance from 2003 on due to the increase in cost of puts with respect to calls after that date. The tail hedging technique presented in the article is fully investable as its turnover is limited; additionally it can replace long/short equity hedge funds for investors who do not have access to alternative investments.
This study highlights the link between stock return volatility, operating performance, and stock returns. Prior studies suggest that there is a ‘low volatility’ anomaly, where firms with a low stock return volatility out-perform firms with a high stock return volatility. This paper confirms that low volatility stocks earn higher returns than high volatility stocks in emerging markets and developed markets outside of North America. We also show that low volatility stocks have higher operating returns and this might explain why low volatility stocks earn higher stock returns. These results provide a partial explanation for the ‘low volatility effect’ that is independent from the existence of market anomalies or per se inefficiencies that might otherwise drive a low volatility effect. We emphasize the importance of controlling for stock return volatility when analyzing operating performance and stock performance.
The authors gauged the return-generating potential of four investment strategies: value weighted, 60/40 fixed mix, and unlevered and levered risk parity. They report three main findings: (1) Even over periods lasting decades, the start and end dates of a backtest can have a material effect on results; (2) transaction costs can reverse ranking, especially if leverage is used; and (3) a statistically significant return premium does not guarantee outperformance over reasonable investment horizons.
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Risk-only investment strategies have been growing in popularity as
traditional in- vestment strategies have fallen short of return targets over
the last decade. However, risk-based investors should be aware of four things.
First, theoretical considerations and empirical studies show that apparently
dictinct risk-based investment strategies are manifestations of a single
effect. Second, turnover and associated transaction costs can be a substantial
drag on return. Third, capital diversification benefits may be reduced. Fourth,
there is an apparent connection between performance and risk diversification.
To analyze risk diversification benefits in a consistent way, we introduce the
Risk Diversification Index (RDI) which measures risk concentrations and
complements the Herfindahl-Herschman Index (HHI) for capital concentrations.
Analytic solutions to Risk Parity, Maximum Diversification, and Minimum Variance portfolios provide useful perspectives about their construction and composition. Individual asset weights depend on both systematic and idiosyncratic risk in all three risk-based portfolios, but systematic risk eliminates many investable assets in long-only constrained Maximum Diversification and Minimum Variance portfolios. On the other hand, all investable assets are included in Risk Parity portfolios, and idiosyncratic risk has little impact on the magnitude of the weights. The algebraic forms for optimal asset weights derived in this paper yield generalizable properties of risk-based portfolios, in contrast to empirical simulations that employ a specific set of historical returns, proprietary risk models, and multiple constraints. The analytic solutions reveal precisely how the various kinds of predicted risk impact the relative magnitude of security weights under each type of risk-based portfolio construction.
We considered five risk-based strategies: equally-weighted, equal-risk budget, equal-risk contribution, minimum variance and maximum diversification. All five can be well described by exposure to the market-cap index and to four simple factors: low-beta, small-cap, low-residual volatility and value. This is, in our view, a major contribution to the understanding of such strategies and provides a simple framework to compare them. All except equally-weighted are defensive with lower volatility than the market-cap index. Equally-weighted is exposed to small-cap stocks. Equal-risk budget and equal-risk contribution are exposed to small-cap and to low-beta stocks. These three have a high correlation of excess returns and their portfolio largely overlap. They invest in all stocks available and have both a low turnover and low tracking error relative to market-cap index. Minimum variance and maximum diversification are essentially exposed to low-beta stocks. They are the most defensive, invest in much the same stocks and have high tracking error and turnover.
Most analysis of risk parity treats it as a heuristic and compares backtests of different allocation methods with less emphasis on investment rationale. The authors investigate risk parity under different settings, highlight its potential utility, and give insights into when this method may be expected to perform better by conducting path independent controlled simulation experiments.
We study the consistency of sample mean-variance portfolios of arbitrarily
high dimension that are based on Bayesian or shrinkage estimation of the input
parameters as well as weighted sampling. In an asymptotic setting where the
number of assets remains comparable in magnitude to the sample size, we provide
a characterization of the estimation risk by providing deterministic
equivalents of the portfolio out-of-sample performance in terms of the
underlying investment scenario. The previous estimates represent a means of
quantifying the amount of risk underestimation and return overestimation of
improved portfolio constructions beyond standard ones. Well-known for the
latter, if not corrected, these deviations lead to inaccurate and overly
optimistic Sharpe-based investment decisions. Our results are based on recent
contributions in the field of random matrix theory. Along with the asymptotic
analysis, the analytical framework allows us to find bias corrections improving
on the achieved out-of-sample performance of typical portfolio constructions.
Some numerical simulations validate our theoretical findings.
One of the ongoing debates in equity market research is the set of common factors that explains the cross section of individual stock returns. With the influential backing of Fama and French [1993], a three-factor model that includes the market, size, and value factors is frequently cited in academic research and widely used in portfolio management. More recently, momentum has joined the list of accepted factors, resulting in references to a four-factor model. Lately, security volatility has begun to be used, along with the factors just mentioned, in describing portfolio risk. The authors introduce a specific measure of the idiosyncratic volatility factor that mirrors the Fama-French methodology, calling it VMS for volatile-minus-stable stocks. VMS is calculated for the entire span of the CRSP database and found to have strong credentials. VMS seems to be more important than SMB (small-minus-big market capitalization) and HML (high-minus-low book-to-market ratio), and similar to UMD (up-minus-down past return) in explaining the covariance structure of stock returns. The relative importance of VMS holds over the entire history for which it can be measured in the U.S. market (1931-2008) and continues to be an important factor in the covariance structure of stock returns in recent decades (1983-2008). Volatility, however, is not very orthogonal to the more well-known factors, a desirable property for new factors. Specifically, VMS is highly correlated to the general market (e.g., volatile stocks outperform stable stocks when the general equity market goes up) despite the fact that the authors measure security volatility in a market-idiosyncratic setting. VMS is also positively correlated with SMB (e.g., volatile stocks tend to outperform when small-cap stocks outperform) despite the Fama-French process of double sorting on market capitalization. Finally, VMS is negatively correlated with HML (e.g., volatile stocks tend to outperform when growth stocks outperform) although this correlation was not pronounced until the last few decades. In contrast to the other Fama-French factors, the average return of the VMS factor has been close to zero over time and negative in recent decades.
ABSTRACT Two easily measured variables, size and book-to-market equity, combine to capture the cross-sectional variation in average stock returns associated with market {3, size, leverage, book-to-market equity, and earnings-price ratios. Moreover, when the tests allow for variation in {3 that is unrelated to size, the relation between market {3 and average return is flat, even when {3 is the only explanatory variable. THE ASSET-PRICING MODEL OF Sharpe (1964), Lintner (1965), and Black (1972)
Considerable attention has recently been given to general equilibrium models of the pricing of capital assets. Of these, perhaps the best known is the mean-variance formulation originally developed by Sharpe (1964) and Treynor (1961), and extended and clarified by Lintner (1965a; 1965b), Mossin (1966), Fama (1968a; 1968b), and Long (1972). In addition Treynor (1965), Sharpe (1966), and Jensen (1968; 1969) have developed portfolio evaluation models which are either based on this asset pricing model or bear a close relation to it. In the development of the asset pricing model it is assumed that (1) all investors are single period risk-averse utility of terminal wealth maximizers and can choose among portfolios solely on the basis of mean and variance, (2) there are no taxes or transactions costs, (3) all investors have homogeneous views regarding the parameters of the joint probability distribution of all security returns, and (4) all investors can borrow and lend at a given riskless rate of interest. The main result of the model is a statement of the relation between the expected risk premiums on individual assets and their "systematic risk." Our main purpose is to present some additional tests of this asset pricing model which avoid some of the problems of earlier studies and which, we believe, provide additional insights into the nature of the structure of security returns. The evidence presented in Section II indicates the expected excess return on an asset is not strictly proportional to its B, and we believe that this evidence, coupled with that given in Section IV, is sufficiently strong to warrant rejection of the traditional form of the model given by (1). We then show in Section III how the cross-sectional tests are subject to measurement error bias, provide a solution to this problem through grouping procedures, and show how cross-sectional methods are relevant to testing the expanded two-factor form of the model. We show in Section IV that the mean of the beta factor has had a positive trend over the period 1931-65 and was on the order of 1.0 to 1.3% per month in the two sample intervals we examined in the period 1948-65. This seems to have been significantly different from the average risk-free rate and indeed is roughly the same size as the average market return of 1.3 and 1.2% per month over the two sample intervals in this period. This evidence seems to be sufficiently strong enough to warrant rejection of the traditional form of the model given by (1). In addition, the standard deviation of the beta factor over these two sample intervals was 2.0 and 2.2% per month, as compared with the standard deviation of the market factor of 3.6 and 3.8% per month. Thus the beta factor seems to be an important determinant of security returns.
In the minimum-variance portfolio, far to the left on the efficient frontier, security weights are independent of expected security returns. Portfolios can be constructed using only the estimated security covariance matrix, without reference to equilibrium expected or actively forecasted returns. Empirical results illustrate the practical value of large-scale numerical optimizations using return-based covariance matrix estimation methodologies, providing new perspective on the factor characteristics of low-volatility portfolios. Optimizations that go back to 1968 reveal that the long-only minimum-variance portfolio has about three-fourths the realized risk of the capitalization-weighted market portfolio, with higher average returns.
This paper describes the advantages of using a particular model of the relationships among securities for practical applications of the Markowitz portfolio analysis technique. A computer program has been developed to take full advantage of the model: 2,000 securities can be analyzed at an extremely low cost--as little as 2% of that associated with standard quadratic programming codes. Moreover, preliminary evidence suggests that the relatively few parameters used by the model can lead to very nearly the same results obtained with much larger sets of relationships among securities. The possibility of low-cost analysis, coupled with a likelihood that a relatively small amount of information need be sacrificed make the model an attractive candidate for initial practical applications of the Markowitz technique.
Stocks with recent past high idiosyncratic volatility have low future average returns around the world. Across 23 developed markets, the difference in average returns between the extreme quintile portfolios sorted on idiosyncratic volatility is -1.31% per month, after controlling for world market, size, and value factors. The effect is individually significant in each G7 country. In the United States, we rule out explanations based on trading frictions, information dissemination, and higher moments. There is strong covariation in the low returns to high-idiosyncratic-volatility stocks across countries, suggesting that broad, not easily diversifiable factors lie behind this phenomenon.
The central message of this paper is that nobody should be using the sample covariance matrix for the purpose of portfolio optimization. It contains estimation error of the kind most likely to perturb a mean-variance optimizer. In its place, we suggest using the matrix obtained from the sample covariance matrix through a transformation called shrinkage. This tends to pull the most extreme coefficients towards more central values, thereby systematically reducing estimation error where it matters most. Statistically, the challenge is to know the optimal shrinkage intensity, and we give the formula for that. Without changing any other step in the portfolio optimization process, we show on actual stock market data that shrinkage reduces tracking error relative to a benchmark index, and substantially increases the realized information ratio of the active portfolio manager.
We evaluate the performance of models for the covariance structure of stock returns, focusing on their use for optimal portfolio
selection. We compare the models' forecasts of future covariances and the optimized portfolios' out-of-sample performance.
A few factors capture the general covariance structure. Portfolio optimization helps for risk control, and a three-factor
model is adequate for selecting the minimum-variance portfolio. Under a tracking error volatility criterion, which is widely
used in practice, larger differences emerge across the models. In general more factors are necessary when the objective is
to minimize tracking error volatility.
We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, "Journal of Financial Economics" 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility. Copyright 2006 by The American Finance Association.
BAYESIAN DECISION THEORY provides formal procedures which utilize information available prior to sampling, together with the sample information, to construct estimates which are optimal with respect to the minimization of the expected loss. This paper presents a method for generating Bayesian estimates of the regression coefficient of rates of return of a security against those of a market index. The distribution of the regression coefficients across securities is used as the prior distribution in the analysis. Explicit formulas are given for the estimates. The Bayesian approach is discussed in comparison with the current practice of sampling-theory procedures.
The Stability of Out-Input Matrices
- M Woodbury
Woodbury, M. The Stability of Out-Input Matrices. (1949) Chicago, Ill.